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Integrability for the Jacobian of orientation preserving mappings. (English) Zbl 0847.26012
The following theorem is proved: Let $$\Omega\subset \mathbb{R}^n$$ be a bounded domain, let $$f: \Omega\to \mathbb{R}^n$$, $$n\geq 2$$, and let $$Jf(x)$$ be the Jacobian of the mapping $$f$$, i.e., $$Jf(x):= \text{det } Df(x)$$. Assume that $$Jf\geq 0$$ and $$f\in L^n (\log L)^{- s}(\Omega)$$, i.e., $$\int_\Omega |Df(x)|^n[\log(1+ |Df(x)|)]^{- s} dx< \infty$$, $$0\leq s\leq 1$$; then $$Jf\in L^1(\log L)^{1- s}_{\text{loc}}$$, i.e., $\int_K Jf(x)[\log(1+ Jf(x))]^{1-s} dx< \infty$ for any compact subset $$K$$ of $$\Omega$$. The theorem fills the gap between S. Müller’s result [J. Reine Angew. Math. 412, 20-34 (1990; Zbl 0713.49004)] obtained for $$s= 0$$ and a result of T. Iwaniec and C. Sbordone [Arch. Ration. Mech. Anal. 119, No. 2, 129-143 (1992; Zbl 0766.46016)] obtained for $$s= 1$$. Moreover, it is shown that the conjecture $$|Df|\in$$ Lorentz space $$L^{n, q}(\Omega)$$, $$(n< q< \infty)\Rightarrow Jf\in L(\log L)^{n/q}_{\text{loc}}(\Omega)$$ is false.

MSC:
 26B10 Implicit function theorems, Jacobians, transformations with several variables 49Q15 Geometric measure and integration theory, integral and normal currents in optimization
Citations:
Zbl 0713.49004; Zbl 0766.46016
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